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© DMTI (2018) | Resource Materials
The Developing Mathematical Thinking Institute (DMTI) is dedicated to enhancing students’ learning of mathematics by supporting educators in the implementation of research-based instructional strategies through high-quality professional development. For more information contact Dr. Brendefur at © DMTI (2018) | Resource Materials
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© DMTI (2018) | Resource Materials
DMTI Warm Ups Warm up Tasks and mini-Lessons Building flexible thinking, fluency and relational thinking © DMTI (2018) | Resource Materials
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© DMTI (2018) | Resource Materials
Warm Ups K-2 Counting to Build One-to-One Correspondence Counting Using Body Movement Walking Paths Counting Using Hand Measurements Skip Counting Challenge Game Quick Draws Shape Drawing 3-6 Skip Counting Challenge Game Counting Using Hand Measurements Quick Draws Shape Drawing Angles with Body Movements Walking Paths © DMTI (2018) | Resource Materials
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© DMTI (2018) | Resource Materials
K-2 Warm Ups © DMTI (2018) | Resource Materials
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Counting to Build One-to-One Correspondence
Draw a row of 8-15 circles on the board. Give the directions to students: “To play this game, you need to count each circle as soon as I touch the board with my counting finger. I will slow down and speed up my counting. You need to follow the way I count with my finger and not count until I touch the next circle.” If students count before you actually touch the board, start counting again from the beginning. © DMTI (2018) | Resource Materials
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Counting Using Body Movement
Students should stand straight and tall. Give the directions to students: “The bottom of your feet, where they touch the ground is 0. If you count all the way to the top of your head you will count to 10. You are 10 units tall. How tall are you? Yes, 10.” Then, squat* slightly lower than the height of 10 and ask students what number you are showing. This should be 9, but many may need clarification of this concept. Then have students start at 10 and gradually count down to 5 (full squat) while they subsequently lower their body to the appropriate height. Finally, give them random numbers from 5 to 11 (e.g. “Show me 7.”) and ask them to show you that height with their bodies. For 11, students may determine they can stand on their toes. *For teachers who may be unable to use their bodies to model the counting, hand gestures can also be utilized. Simply give students a height to try to match with their bodies with a raised hand and then move that hand up and down to help guide the students as to how they should move their bodies. © DMTI (2018) | Resource Materials
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© DMTI (2018) | Resource Materials
Walking Paths 1. Have 3-4 students stand next to you in a line with all of your backs up against a wall or specific landmark in the room. 2. Tell the class that they will be counting how many steps away from the landmark you and the students are. 3. Ask the class, “If we haven’t taken any steps, what number should we say that means we haven’t gone anywhere yet? What number means nothing?” (Answer: 0). 4. Students should be guided to count the steps away from the landmark you and the 3-4 students take only as your feet touch the ground. Guide the 3-4 students with you by saying, “Step forward,” quietly for each step. 5. After 4-6 steps have been taken, have the students walking with you step backward. Guide the class to recognize that if you were 5 steps away from the landmark and stepped backwards, you are now 4 steps away. Continue this process until you arrive at the starting location, which would be 0. 6. Repeat this process in small groups and then have students do this in small groups without direction once they are familiar with the directions for the activity. © DMTI (2018) | Resource Materials
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Counting Using Hand Measurements
1. Place your hands or index fingers together. Ask students to think of a number that could be used to name how far apart your fingers/hands are if there is no space between them. (Answer: 0) 2. Have students count aloud and move their fingers/hands in unison with you as you count to 10 and show the corresponding lengths with the space between your fingers/hands. 3. Then count backwards from 10 to 0 following the same process. 4. Finally, have students show certain lengths with their fingers/hands (e.g. “Show me 7. Show me 4. What is 1 more than 4? Yes, 5. Now show me 6. Show me 7. Show me 8. What comes next? Yes, 9. Show me 10. Now show me 5. Now 10. Now 5. Now 4, 5, 6, 4, 5, 6, 7, 8, 9, 10, 5, 0.” © DMTI (2018) | Resource Materials
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Skip Counting Challenge Game
1. Choose an easy number to count by (e.g. 1, 10 or 5) and have students count up and back down using that number. 2. Tell students that you will point to yourself and count and that they cannot count out loud when you are pointing to yourself but that when you point to them, they should continue to count on from where you stopped and that you cannot count when you are pointing to them. 3. Practice this process by alternating the counting sequence starting at 0. 4. The goal of the Skip Counting Challenge Game is for the students to follow your counting without making a mistake (e.g. counting out loud when you’re not pointing to them) to a given target number. For example, counting by 5’s from 0 to 100. Be sure to vary the pattern of where you are pointing so that students must attend to where you are counting and think of the counting sequence without speaking out loud. © DMTI (2018) | Resource Materials
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© DMTI (2018) | Resource Materials
Quick Draw - Number Draw this model. If this is a model of 1, draw a model of what 2 would look like. 1 Students may show each individual unit of 1 inside their model for 2. Challenge students to draw the 2 using both methods. 2 © DMTI (2018) | Resource Materials
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© DMTI (2018) | Resource Materials
Quick Draw - Number Draw this model. If this is a model of 2, draw a model of what 1 would look like. 2 1 © DMTI (2018) | Resource Materials
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© DMTI (2018) | Resource Materials
Quick Draw - Number Draw this model. If this is a model of 1, draw a model of what 3 would look like. What would 6 look like? 1 Students may show each individual unit of 1 inside their model for 3. Challenge students to draw the 3 using both methods. The same may be true with their models for 6 and if so, follow the same process. 3 6 © DMTI (2018) | Resource Materials
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© DMTI (2018) | Resource Materials
Quick Draw - Number Continue the same process with various numbers. Alternate between giving students a unit that needs to be iterated to compose a larger unit and then a unit that must be partitioned/decomposed to create a smaller unit. Special emphasis should be given to: - Flexible sizes and shapes of units of place value (e.g. 1 and 10) - Students using the terms partition, iterate, compose, decompose and unit to describe their drawing strategies © DMTI (2018) | Resource Materials
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© DMTI (2018) | Resource Materials
Quick Draw - Shape Have students draw the following shape. Have them focus on the lengths and proportions. Have them edit their work if it is not correct. Then, have them imagine the shape turned 90 or degrees to the right or to the left and draw that shape. You do not need to use the terms “90 degrees” or “180 degrees”. Instead, you may choose to guide students to use the sides or the given letters to match the rotation path. For example say, “Turn the shape so that the letter is right here.” The next slides provide a few examples, but look at the Quick Draw – Shapes for more examples. © DMTI (2018) | Resource Materials
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© DMTI (2018) | Resource Materials
Quick Draw - Shape Draw the following shape as precisely as possible. Now turn the shape 90⁰ to the right. © DMTI (2018) | Resource Materials
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© DMTI (2018) | Resource Materials
Quick Draw - Shape Draw the turned shape as precisely as possible. © DMTI (2018) | Resource Materials
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© DMTI (2018) | Resource Materials
Shape Drawing Have students use their index fingers to draw shapes in the air. Have students turn and move the shapes and ask them to state that the shape is still a (circle, rectangle, square, triangle, etc.) even though it is in different location or different orientation. Then have students physically “stretch” or “squish” the shape to transform its shape. For example, changing a square into a rectangle (that is not a square) or changing a circle into an oval. Students should describe how they changed the shape and what was the same and different between the two shapes. Have students use appropriate academic language (e.g. sides, vertices) to describe the shapes. © DMTI (2018) | Resource Materials
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© DMTI (2018) | Resource Materials
3-6 Warm Ups © DMTI (2018) | Resource Materials
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Skip Counting Challenge Game
1. Choose any number to count by (whole numbers or fractions/decimals) and have students count up and back down using that number. 2. Tell students that you will point to yourself and count and that they cannot count out loud when you are pointing to yourself but that when you point to them, they should continue to count on from where you stopped and that you cannot count when you are pointing to them. 3. Practice this process by alternating the counting sequence starting at 0. 4. The goal of the Skip Counting Challenge Game is for the students to follow your counting without making a mistake (e.g. counting out loud when you’re not pointing to them) to a given target number. For example, counting by units of from 0 to 5. Be sure to vary the pattern of where you are pointing so that students must attend to where you are counting and think of the counting sequence without speaking out loud. If you are counting by fractions, students can be asked to signal whole number equivalents (e.g =3) with hand gestures such as “thumbs up” or raised hands. © DMTI (2018) | Resource Materials
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Counting Using Hand Measurements
1. Place your hands or index fingers together. Ask students to think of a number that could be used to name how far apart your fingers/hands are if there is no space between them. (Answer: 0) 2. Have students count aloud and move their fingers/hands in unison with you as you count by any number (e.g. 4, 12, .5, 1 3 ) to a reasonable stopping point. 3. Then count backwards from the ending number to 0 following the same process. 4. Try to have students show various reference points or benchmarks in their counting and hand gestures (e.g. 0, 50, 100 if counting by 10’s). © DMTI (2018) | Resource Materials
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© DMTI (2018) | Resource Materials
Quick Draws Draw this model. If this is a model of 4, draw a model of what 8 would look like. 4 Students may show each individual unit of 4 inside their model for 8. Challenge students to draw the 8 using both methods. 8 © DMTI (2018) | Resource Materials
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© DMTI (2018) | Resource Materials
Quick Draws Draw this model. If this is a model of 12, draw a model of what 3 would look like. 12 3 © DMTI (2018) | Resource Materials
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© DMTI (2018) | Resource Materials
Quick Draws Draw this model. If this is a model of 5, draw a model of what 15 would look like. What would 30 look like? 5 Students may show each individual unit of 4 inside their model for 12. Challenge students to draw the 15 using both methods. The same may be true with their models for 30 and if so, follow the same process. 15 30 © DMTI (2018) | Resource Materials
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© DMTI (2018) | Resource Materials
Quick Draws Continue the same process with various numbers. Alternate between giving students a unit that needs to be iterated to compose a larger unit and then a unit that must be partitioned/decomposed to create a smaller unit. Special emphasis should be given to: - Flexible sizes and shapes of units and especially if using fractions. - Students using the terms partition, iterate, compose, decompose and unit to describe their drawing strategies. © DMTI (2018) | Resource Materials
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© DMTI (2018) | Resource Materials
Shape Drawing Have students use their index fingers to draw shapes in the air. Have students turn and move the shapes and ask them to state that the shape is still a (circle, rectangle, square, triangle, etc.) even though it is in different location or different orientation. Then have students physically “stretch” or “squish” the shape to transform its shape. For example, changing a square into a rectangle (that is not a square) or changing a circle into an oval. Students should describe how they changed the shape and what was the same and different between the two shapes. Have students use appropriate academic language (e.g. sides, vertices) to describe the shapes. Challenge students to decompose or partition the drawn shapes and to describe what new shapes are created. © DMTI (2018) | Resource Materials
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Angles with Body Movements
1. Have students stand up and put one are straight in front of them. This location represents 00. 2. Have students rotate their arm and body clockwise to show movements modeling benchmark angels (e.g. 450, 900, 1800, 3600). 3. Challenge students to move in ways that model less familiar angle measurements such as or 750. Students should use (and verbally describe) relational thinking strategies to determine what these angle measurements should look like. 4. Finally, have students put both arms in front of them and keep their left arm stationary as they sweep their right arm clockwise to show a variety of angle measurements. This modification of the task will connect to the more commonplace angle measurements students will use to describe polygons and two-dimensional figures because they are measuring the degrees that one segment (arm) sweeps from another. © DMTI (2018) | Resource Materials
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© DMTI (2018) | Resource Materials
Walking Paths 1. Have 3-4 students stand next to you in a line with all of your backs up against a wall or specific landmark in the room. 2. Tell the class that they will be counting how many steps away from the landmark you and the students are. 3. Ask the class, “If we haven’t taken any steps, what number should we say that means we haven’t gone anywhere yet? What number means nothing?” (Answer: 0). 4. Students should be guided to count the steps away from the landmark you and the 3-4 students take only as your feet touch the ground. Guide the 3-4 students with you by saying, “Step forward,” quietly for each step. As a class, agree on the unit that each step will represent. These units could be whole numbers, decimals or fractions. 5. After several steps have been taken, have the students walking with you step backward. Guide the class to recognize that they are now counting backwards as you are measuring the distance you are from the landmark. Continue this process until you arrive at the starting location, which would be 0. 6. Repeat this process in small groups and then have students do this in small groups without direction once they are familiar with the directions for the activity. © DMTI (2018) | Resource Materials
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© DMTI (2018) | Resource Materials
Brendefur and Strother (2018). DMTI Inc. © DMTI (2018) | Resource Materials
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